Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Examinations — 16-Mec-A6 Fluid Machinery, May 2019. Closed-book, three hours. Section A is calculative (5 questions) and Section B is descriptive (3 questions); candidates answer four questions from Section A and two from Section B (six questions, 60 marks, 10 marks each). All eight questions are solved as a study resource.
Reference texts. Dixon & Hall, Fluid Mechanics and Thermodynamics of Turbomachinery (7th ed.); Cohen, Rogers & Saravanamuttoo, Gas Turbine Theory (6th ed.); Fox & McDonald, Introduction to Fluid Mechanics (10th ed.); F. M. White, Fluid Mechanics (8th ed.); Çengel & Boles, Thermodynamics: An Engineering Approach (9th ed.).
Paper. This is the 16-Mec-A6, May 2019 Fluid Machinery examination.
Given. Axial compressor first stage: hub $D_1=0.480\;\text{m}$, tip $D_2=1.120\;\text{m}$, absolute inlet flow angle $\alpha_1=30^\circ$ (from axial), relative outlet angle $\beta_2=40^\circ$ (from axial), $N=6800\;\text{rev/min}$, $\dot m=136\;\text{kg/s}$, $T_1=288\;\text{K}$; air $c_p=1005\;\text{J/kg\,K}$, $\gamma=1.4$, $R=287\;\text{J/kg\,K}$.
Given data
Hub $D_1$
Tip $D_2$
$\alpha_1$
$\beta_2$
$N$
$\dot m$
$T_1$
0.480 m
1.120 m
30°
40°
6800 rpm
136 kg/s
288 K
Find. Mean blade speed and axial velocity, the four velocities from the triangles, the stage work and power, the temperature rise, and the isentropic pressure ratio.
Approach. Mean blade speed from the mean diameter; axial velocity from continuity through the inlet annulus; close the velocity triangles with axial velocity assumed constant; get work from Euler's equation $w=U\,\Delta C_w$, temperature rise from $w=c_p\Delta T_0$, and the pressure ratio from the isentropic relation.
Combined first-stage velocity triangle on the common blade-speed base $U$. Absolute velocities $C$ (from the fixed frame) and relative velocities $W$ (from the rotor) share the axial component $C_x$; the whirl grows from inlet to outlet, and that change $\Delta C_w$ drives the work.
Check: inlet pressure is not stated; standard sea-level ambient $p_1=101.3\;\text{kPa}$ is assumed for the inlet density ($\rho_1=1.226\;\text{kg/m}^3$). This sets the axial velocity through continuity.
Mean blade speed. Mean diameter $D=\tfrac{1}{2}(D_1+D_2)=0.800\;\text{m}$, so
$$U=\pi D\,\frac{N}{60}=\pi(0.800)\frac{6800}{60}=\boxed{284.9\;\text{m/s}}$$
Axial velocity from continuity. Annulus area $A=\tfrac{\pi}{4}(D_2^2-D_1^2)=\tfrac{\pi}{4}(1.120^2-0.480^2)=0.804\;\text{m}^2$; with $\rho_1=p_1/RT_1=1.226\;\text{kg/m}^3$,
$$C_{x1}=\frac{\dot m}{\rho_1 A}=\frac{136}{1.226\times0.804}=\boxed{138.0\;\text{m/s}}$$
Inlet triangle. The absolute flow leaves the guide vanes at $\alpha_1=30^\circ$, so the inlet whirl is $C_{w1}=C_{x1}\tan\alpha_1=138.0\tan30^\circ=79.7\;\text{m/s}$ and $C_1=C_{x1}/\cos\alpha_1=159.3\;\text{m/s}$. The relative velocity closes on $U$: $W_{w1}=U-C_{w1}=205.2\;\text{m/s}$, giving
$$W_1=\sqrt{C_{x1}^2+W_{w1}^2}=247.3\;\text{m/s},\qquad \beta_1=\arctan\frac{W_{w1}}{C_{x1}}=56.1^\circ.$$
Outlet triangle. With axial velocity constant and $\beta_2=40^\circ$, the exit relative whirl is $W_{w2}=C_{x1}\tan\beta_2=115.8\;\text{m/s}$, so the absolute exit whirl is $C_{w2}=U-W_{w2}=169.1\;\text{m/s}$, and
$$W_2=\frac{C_{x1}}{\cos\beta_2}=180.1\;\text{m/s},\qquad C_2=\sqrt{C_{x1}^2+C_{w2}^2}=218.2\;\text{m/s}.$$
Work and power (Euler). With no inlet whirl correction beyond the vanes, $\Delta C_w=C_{w2}-C_{w1}=89.4\;\text{m/s}$, hence
$$w=U\,\Delta C_w=284.9\times89.4=\boxed{25.5\;\text{kJ/kg}},\quad \dot W=\dot m\,w=136\times25.5\times10^{3}=3.47\times10^{6}\;\text{W}\;(3465\;\text{kW}).$$
Temperature rise. All the shaft work raises the stagnation enthalpy, $w=c_p\,\Delta T_0$, so
$$\Delta T_0=\frac{w}{c_p}=\frac{25\,480}{1005}=\boxed{25.4\;\text{K}}$$ (the enthalpy rise is $\Delta h_0=25.5\;\text{kJ/kg}$).
Isentropic pressure ratio. For an ideal stage
$$\frac{p_{02}}{p_{01}}=\left(1+\frac{\Delta T_0}{T_1}\right)^{\gamma/(\gamma-1)}=\left(1+\frac{25.4}{288}\right)^{3.5}=\boxed{1.343}$$
